# Span (linear algebra)

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## Definition

Let [ilmath](V,\mathbb{F})[/ilmath] be a vector space over a field [ilmath]\mathbb{F} [/ilmath] and let [ilmath]\{v_\alpha\}_{\alpha\in I}\subseteq V[/ilmath] be an arbitrary collection of vectors of [ilmath]V[/ilmath]. The span of [ilmath]\{v_\alpha\}_{\alpha\in I} [/ilmath], denoted:

• [ilmath]\text{Span}(\{v_\alpha\}_{\alpha\in I})[/ilmath] for an arbitrary collection, or
• [ilmath]\text{Span}(v_1,\ldots,v_k)[/ilmath] for a finite collection

is defined as follows:

• $\text{Span}(\{v_\alpha\}_{\alpha\in I}):\eq\Bigg\{\underbrace{\sum_{\alpha\in I}\lambda_\alpha v_\alpha}_{\text{Linear combination} }\ \Bigg\vert\ \{\lambda_\alpha\}_{\alpha\in I}\in\underbrace{\big\{\{\lambda_\alpha\}_{\alpha\in I}\in\mathbb{F}^I\ \big\vert\ \overbrace{\vert\{\lambda_\alpha\ \vert\ \alpha\in I\wedge\lambda_\alpha\neq 0\}\vert\in\mathbb{N} }^{\text{There are only finitely many non-zero terms} }\big\} }_{\text{The set of }I\text{-indexed scalars such that}\{\lambda_\alpha\}_{\alpha\in I}\text{ only has finitely many non-zero terms} } \Bigg\}$[Note 1]

For a finite collection, [ilmath]\{v_1,\ldots,v_k\} [/ilmath] this simplifies to:

• [ilmath]\text{Span}(v_1,\ldots,v_n):\eq\left\{\left.\sum_{i\eq 1}^k\lambda_i v_i\ \right\vert\ (\lambda_i)_{i\eq 1}^k\in\mathbb{F}\right\} [/ilmath]

See linear combination for details of why we need the "finitely many part"

### Caveats

Caveat:There are a few problems here

1. Ordered basis - in the set [ilmath]\{v_1,\ldots,v_n\} [/ilmath] there is no order, we really mean [ilmath](v_i)_{i\eq 1}^n[/ilmath] - this implies order. Also for an arbitrary collection, tuple notation doesn't make sense unless there's an ordering in play. This needs to be "united"